Theorem cofu2nd 18065

Description: Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
cofuval.b	⊢ 𝐵 = (Base‘𝐶)
cofuval.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
cofuval.g	⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
cofu2nd.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
cofu2nd.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
cofu2nd	⊢ (𝜑 → (𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌)))

Proof of Theorem cofu2nd

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cofuval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
2		cofuval.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
3		cofuval.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
4	1, 2, 3	cofuval 18062	. . . 4 ⊢ (𝜑 → (𝐺 ∘func 𝐹) = ⟨((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))⟩)
5	4	fveq2d 6889	. . 3 ⊢ (𝜑 → (2nd ‘(𝐺 ∘func 𝐹)) = (2nd ‘⟨((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))⟩))
6		fvex 6898	. . . . 5 ⊢ (1st ‘𝐺) ∈ V
7		fvex 6898	. . . . 5 ⊢ (1st ‘𝐹) ∈ V
8	6, 7	coex 7942	. . . 4 ⊢ ((1st ‘𝐺) ∘ (1st ‘𝐹)) ∈ V
9	1	fvexi 6899	. . . . 5 ⊢ 𝐵 ∈ V
10	9, 9	mpoex 8092	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦))) ∈ V
11	8, 10	op2nd 8010	. . 3 ⊢ (2nd ‘⟨((1st ‘𝐺) ∘ (1st ‘𝐹)), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))⟩) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)))
12	5, 11	eqtrdi 2811	. 2 ⊢ (𝜑 → (2nd ‘(𝐺 ∘func 𝐹)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦))))
13		simprl 783	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑥 = 𝑋)
14	13	fveq2d 6889	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((1st ‘𝐹)‘𝑥) = ((1st ‘𝐹)‘𝑋))
15		simprr 785	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
16	15	fveq2d 6889	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((1st ‘𝐹)‘𝑦) = ((1st ‘𝐹)‘𝑌))
17	14, 16	oveq12d 7438	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) = (((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)))
18	13, 15	oveq12d 7438	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥(2nd ‘𝐹)𝑦) = (𝑋(2nd ‘𝐹)𝑌))
19	17, 18	coeq12d 5841	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((((1st ‘𝐹)‘𝑥)(2nd ‘𝐺)((1st ‘𝐹)‘𝑦)) ∘ (𝑥(2nd ‘𝐹)𝑦)) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌)))
20		cofu2nd.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
21		cofu2nd.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
22		ovex 7453	. . . 4 ⊢ (((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∈ V
23		ovex 7453	. . . 4 ⊢ (𝑋(2nd ‘𝐹)𝑌) ∈ V
24	22, 23	coex 7942	. . 3 ⊢ ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌)) ∈ V
25	24	a1i 11	. 2 ⊢ (𝜑 → ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌)) ∈ V)
26	12, 19, 20, 21, 25	ovmpod 7572	1 ⊢ (𝜑 → (𝑋(2nd ‘(𝐺 ∘func 𝐹))𝑌) = ((((1st ‘𝐹)‘𝑋)(2nd ‘𝐺)((1st ‘𝐹)‘𝑌)) ∘ (𝑋(2nd ‘𝐹)𝑌)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ⟨cop 4834   ∘ ccom 5654  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000  Basecbs 17392   Func cfunc 18034   ∘_func ccofu 18036

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-map 8852  df-ixp 8929  df-func 18038  df-cofu 18040

This theorem is used by:  cofu2  18066  cofucl  18068  cofuass  18069  cofull  18116  cofth  18117  catciso  18291  cofidf2a  50326